Bayesian bivariate quantile regression

Quantile regression (QR) has become a widely used tool to study the impact of covariates on quantiles of a response distribution. QR provides a detailed description of the conditional response when considering a dense set of quantiles, without assuming a closed form for its distribution. The Bayesian version of QR, which can be implemented by considering the asymmetric Laplace distribution (ALD) as an auxiliary error distribution, is an attractive alternative to other methods because it returns knowledge on the whole parameter distribution instead of solely point estimations. While for the univariate case there has been a lot of development in the last few years, multivariate responses have only been treated to a little extent in the literature, especially in the Bayesian case. By using a multivariate version of the location scale mixture representation for the ALD, we are able to apply inference techniques developed for multivariate Gaussian models on multivariate quantile regression and make thus the impact of covariates on the quantiles of more than one dependent variable feasible. The model structure also facilitates the determination of conditional correlations between bivariate responses on different quantile levels after adjusting for covariate effects.